Cable Footbridge

What is modelled

A shallow cable carries its own weight and the deck as a uniform load. The horizontal pull is set by the sag: the flatter the cable, the larger the pull. The same pull makes the cable behave like a stretched string, so it vibrates in modes with frequency proportional to the mode number. A crowd stepping at one of those frequencies can make the bridge move a lot, and the wind pushes it sideways and sheds vortices at its own rate.

Equations

• Pull H = w L2 / (8 f), peak tension T = H √(1 + (4f/L)2), with w the weight per metre.
• String modes fn = (n / 2L) √(H / m).
• Resonance q = Q / (k √((1 − r2)2 + (2 ζ r)2)), with r = fstep / fn, modal mass mL/2 and modal force Q = N G α sin(nπ/2) for N walkers at midspan.
• Wind qw = ½ ρ CD d v2, sway qw L2 / (8 H), vortex frequency fs = St v / d.

Checks run in the page

• The parabola against the exact catenary peak tension.
• The cable length by Simpson integration and the support reaction from the end slope.
• The resonance amplitude from RK4 integration against the closed form.
• The mode frequency of a chain of 80 masses found by a Sturm bisection.

Honest simplifications

The deck follows the cable and has no bending stiffness, the walkers are a worst-case synchronised crowd at midspan with a dynamic load factor of 0.4 and a weight of 750 N each, only one mode is driven at a time, the wind coefficients are single textbook values and the sway is a static estimate. Use it for the trends, not for designing a bridge.

• Press Resonant crowd and watch the midspan motion grow when the steps match the mode.
• Move the step frequency away from the mode to see the response fall.
• Press Storm wind and compare the vortex frequency with the mode frequency.